smc

Let the length of the two arcs be denoted $x$. First, we acknowledge that the length of an arc of a circle is given by the measure of that arc's central angle in radians times the radius of that circle. Note that the central angle of circle $I$ measures $\frac{\pi }{3}$ radians, and the central angle of circle $II$ measures $\frac{\pi }{4}$ radians. Let the radius of circle $I$ be $r_1$ and the radius of circle $II$ be $r_2$. Then, we know that the arc length $x=\frac{\pi }{3}{r}_1=\frac{\pi }{4}{r}_2$, and so $\frac{r_1}{r_2}=\frac{3}{4}$. From this, we see that $\frac{r_1^2}{r_2^2}=\frac{\pi r_1^2}{\pi r_2^2}=\frac{[I]}{[II]}=\frac{9}{16}$. Thus, our answer is $\boxed{9:16}$.